130 research outputs found
Revisiting the Equivalence Problem for Finite Multitape Automata
The decidability of determining equivalence of deterministic multitape
automata (or transducers) was a longstanding open problem until it was resolved
by Harju and Karhum\"{a}ki in the early 1990s. Their proof of decidability
yields a co_NP upper bound, but apparently not much more is known about the
complexity of the problem. In this paper we give an alternative proof of
decidability, which follows the basic strategy of Harju and Karhumaki but
replaces their use of group theory with results on matrix algebras. From our
proof we obtain a simple randomised algorithm for deciding language equivalence
of deterministic multitape automata and, more generally, multiplicity
equivalence of nondeterministic multitape automata. The algorithm involves only
matrix exponentiation and runs in polynomial time for each fixed number of
tapes. If the two input automata are inequivalent then the algorithm outputs a
word on which they differ
Revisiting Membership Problems in Subclasses of Rational Relations
We revisit the membership problem for subclasses of rational relations over
finite and infinite words: Given a relation R in a class C_2, does R belong to
a smaller class C_1? The subclasses of rational relations that we consider are
formed by the deterministic rational relations, synchronous (also called
automatic or regular) relations, and recognizable relations. For almost all
versions of the membership problem, determining the precise complexity or even
decidability has remained an open problem for almost two decades. In this
paper, we provide improved complexity and new decidability results. (i) Testing
whether a synchronous relation over infinite words is recognizable is
NL-complete (PSPACE-complete) if the relation is given by a deterministic
(nondeterministic) omega-automaton. This fully settles the complexity of this
recognizability problem, matching the complexity of the same problem over
finite words. (ii) Testing whether a deterministic rational binary relation is
recognizable is decidable in polynomial time, which improves a previously known
double exponential time upper bound. For relations of higher arity, we present
a randomized exponential time algorithm. (iii) We provide the first algorithm
to decide whether a deterministic rational relation is synchronous. For binary
relations the algorithm even runs in polynomial time
On Minimality and Size Reduction of One-Tape and Multitape Finite Automata
In this thesis, we consider minimality and size reduction issues of one-tape and multitape automata. Although the topic of minimization of one-tape automata has been widely studied for many years, it seems that some issues have not gained attention. One of these issues concerns finding specific conditions on automata that imply their minimality in the class of nondeterministic finite automata (NFA) accepting the same language. Using the theory of NFA minimization developed by Kameda and Weiner in 1970, we show that any bideterministic automaton (that is, a deterministic automaton with its reversal also being deterministic) is a unique minimal automaton among all NFA accepting its language. In addition to the minimality in regard to the number of states, we also show its minimality in the number of transitions. Using the same theory of Kameda and Weiner, we also obtain a more general minimality result. We specify a set of sufficient conditions under which a minimal deterministic automaton (DFA
On equivalence, languages equivalence and minimization of multi-letter and multi-letter measure-many quantum automata
We first show that given a -letter quantum finite automata
and a -letter quantum finite automata over
the same input alphabet , they are equivalent if and only if they are
-equivalent where , , are the
numbers of state in respectively, and . By
applying a method, due to the author, used to deal with the equivalence problem
of {\it measure many one-way quantum finite automata}, we also show that a
-letter measure many quantum finite automaton and a
-letter measure many quantum finite automaton are
equivalent if and only if they are -equivalent
where , , are the numbers of state in respectively,
and .
Next, we study the language equivalence problem of those two kinds of quantum
finite automata. We show that for -letter quantum finite automata, the
non-strict cut-point language equivalence problem is undecidable, i.e., it is
undecidable whether
where
and are -letter quantum finite automata.
Further, we show that both strict and non-strict cut-point language equivalence
problem for -letter measure many quantum finite automata are undecidable.
The direct consequences of the above outcomes are summarized in the paper.
Finally, we comment on existing proofs about the minimization problem of one
way quantum finite automata not only because we have been showing great
interest in this kind of problem, which is very important in classical automata
theory, but also due to that the problem itself, personally, is a challenge.
This problem actually remains open.Comment: 30 pages, conclusion section correcte
On Rational Recursive Sequences
We study the class of rational recursive sequences (ratrec) over the rational
numbers. A ratrec sequence is defined via a system of sequences using mutually
recursive equations of depth 1, where the next values are computed as rational
functions of the previous values. An alternative class is that of simple ratrec
sequences, where one uses a single recursive equation, however of depth k: the
next value is defined as a rational function of k previous values.
We conjecture that the classes ratrec and simple ratrec coincide. The main
contribution of this paper is a proof of a variant of this conjecture where the
initial conditions are treated symbolically, using a formal variable per
sequence, while the sequences themselves consist of rational functions over
those variables. While the initial conjecture does not follow from this
variant, we hope that the introduced algebraic techniques may eventually be
helpful in resolving the problem.
The class ratrec strictly generalises a well-known class of polynomial
recursive sequences (polyrec). These are defined like ratrec, but using
polynomial functions instead of rational ones. One can observe that if our
conjecture is true and effective, then we can improve the complexities of the
zeroness and the equivalence problems for polyrec sequences. Currently, the
only known upper bound is Ackermanian, which follows from results on polynomial
automata. We complement this observation by proving a PSPACE lower bound for
both problems for polyrec. Our lower bound construction also implies that the
Skolem problem is PSPACE-hard for the polyrec class
Revisiting the Equivalence Problem for Finite Multitape Automata
Abstract. The decidability of determining equivalence of deterministic multitape automata was a longstanding open problem until it was resolved by Harju and Karhumäki in the early 1990s. Their proof of decidability yields a co-NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability which follows the basic strategy of Harju and Karhumäki, but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding equivalence of deterministic multitape automata, as well as automata with transition weights in the field of rational numbers. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ
Unit Resolution for a Subclass of the Ackermann Class
The Ackermann class and the Gödel class are typical subclasses of pure first-order logic. The unsatisfiability problems for the Ackermann class and the Gödel class of formulas are decidable and resolution strategies to the unsatisfiability problems for the Ackermann class and the Gödel class were constructed by W. H. Joyner. Applying unit resolution of C. L. Chang, we construct a preprocessor to Joyner's resolution strategy for a subclass of the Ackermann class, since his strategy may necessitate too much time and space from the practical point of view. In this paper, we describe an algorithm to decide whether there is a unit resolution refutation from a set of clauses in a subclass ACK₂ of the Ackermann class, in which at most two literals with variables appear in each clause. In this algorithm, we represent the unit clause resolvents generated by unit resolution by means of finite automata. Also, we transform the decision problem of a unit resolution refutability for ACK₂ to the emptiness problem of intersections of two regular languages. We give the time complexity and the space complexity of the constructed algorithm. This result is an extension of the result by N. D. Jones namely that it can be decided in deterministic polynomial time whether or not ther is a unit resolution refutation for the propositional logic
Superlinear lower bounds based on ETH
Andras Z. Salamon acknowledges support from EPSRC grants EP/P015638/1 and EP/V027182/1.We introduce techniques for proving superlinear conditional lower bounds for polynomial time problems. In particular, we show that CircuitSAT for circuits with m gates and log(m) inputs (denoted by log-CircuitSAT) is not decidable in essentially-linear time unless the exponential time hypothesis (ETH) is false and k-Clique is decidable in essentially-linear time in terms of the graph's size for all fixed k. Such conditional lower bounds have previously only been demonstrated relative to the strong exponential time hypothesis (SETH). Our results therefore offer significant progress towards proving unconditional s uperlinear time complexity lower bounds for natural problems in polynomial time.Postprin
A SURVEY OF LIMITED NONDETERMINISM IN COMPUTATIONAL COMPLEXITY THEORY
Nondeterminism is typically used as an inherent part of the computational models used incomputational complexity. However, much work has been done looking at nondeterminism asa separate resource added to deterministic machines. This survey examines several differentapproaches to limiting the amount of nondeterminism, including Kintala and Fischer\u27s βhierarchy, and Cai and Chen\u27s guess-and-check model
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